GNSS OBSERVABLES João F. Galera Monico - UNESP Tuesday Sep
Basic references
Basic GNSS Observation Equations Pseudorange Carrier Phase Doppler SNR Signal to Noise Ratio
Pseudorange Observation Equation (P)
P r k i = ρ r k + c dt r dt s + T k k s r + I r i + M Pi + b r,pi b Pi + e i
Carrier-Phase Observation Equation (L)
DCB Differential Code Bias (code) / UPD (uncalibrated phase delays)
Doppler Observation Equation (D) Doppler shift caused by the relative motion of receiver and satellite. Additionally, the receiver s or satellite s clocks may be affected by a frequency offset or drift Only the geometric Doppler effect was considered. Atmospheric propagation delay, clock frequency deviations, and relativistic effects were neglected e is the unit line of-sight vector from the user to the GNSS satellite We assume that vr << c and v s << c, Deviation due to relativistic effects Satellite frequency deviation from standard Receiver frequency deviation from standard
Covariance Matrix of the observations Together with the observation equation, a covariance matrix (CM) of the observations has to be available. In general, no correlations are assumed among the original observables (diagonal matrix). Carrier phase is normally order of magnitude better than Pseudorange.
CM of the observations It is usual to taking into account the satellite elevation angle, CNo, or others to built the CM. Considering Pseudoranges and two carrier phases at f i, and precision as function of the elevation angle The CM may be given as: CM i s pd... s pd...... s L... s L s s L pd pd L *f(elev) pd L *f(elev) * const const for example
CM of the observations............ L L pd pd i s s s s CM example for const const s s L pd pd pd L L * *f(elev) *f(elev) /sin(ele) f(elev) cos(ele) f(elev) Two examples
An example of an RINEX file
Combination of observations Several combinations between the GNSS observables are available. to minimize or better estimate specific quantities. GNSS processing using two or more frequencies is usually accomplished by using combined observables taking advantage of each combination/characteristics.
Combination of observations Combination among observation of different GNSS system
Combination of observations Combination of single satellite and receiver (only code, only carrier or both) Combination of Multisatellite and Multireceiver (only code, only carrier or both)
Combination of single satellite Applications and receiver Carrier-phase ambiguity resolution Isolation or elimination of ionospheric errors Multipath analysis
Wide and Narrow Lane Combinations. In the wide-lane combinations, the combined wavelength is larger than the largest individual wavelength in the combination.. In the intermediate-lane combinations, the combined wavelength lies between the largest and the shortest individual wavelength. 3. The narrow-lane combinations have a shorter wavelength than the individual signal with the shortest wavelength in the combination.
Wide Lane
Wide Lane The large wavelength of the Wide-lane combination (~86 cm) is useful for ambiguity resolution algorithms, as well as cycle-slip and outliers detection. But it is important to emphasize that noises present in the original observables are also amplified.
Narrow Lane
Narrow Lane The NL combinations have lower noise than the original observables used to generate them. The narrow wavelength (~,4) turns ambiguity resolution harder in comparison to other combination.
Ionospheric-free linear combination One of the most important combinations commonly referred as iono-free. This combination considerably removes the first order ionospheric effect, up-to 99.9%, which is frequency dependent. The remaining.% of ionospheric refraction affecting the measurements corresponds to only a few centimeters or even less (MARQUES, MONICO AND AQUINO () GPS Solution 5-3).
Ionospheric-free linear combination Dual Frequency f / ( f f ) f f / ( f f )
Ionospheric-free linear combination Dual Frequency GNSS positioning using code and phase ionospheric free observables the so-called Timing Group Delays (TGDs) are also cancelled in the combination. The satellite clocks errors (by definition) refer to the ionospheric free combination (Laurichesse, 8; Sanz Subirana et al., 3).
Ionospheric-free linear combination Single Frequency GRAPHIC: (GRoup and PHase Ionospheric Calibration) GRAPHIC only requires pseudorange and carrier-phase observations on one frequency. So, it is even suitable for use with inexpensive single-frequency receivers. L N GRAP GRAP N ( P i i / s Li ) c( dr dt ) T i NGRAP W is the (float valued) GRAPHIC ambiguity e GRAP
Ionospheric-free linear combination Single Frequency The DCBs need also be considered when combining the GRAPHIC observations with (single-frequency) pseudoranges to enable estimation of both the receiver clock offset and the GRAPHIC ambiguities. Depending of the requirements, tropospheric delays in may be considered.
Except for very short data arcs that do not enable proper estimation of the ambiguities, the GRAPHIC based PPP solution usually offers better positioning results than single-frequency pseudorange processing with a priori corrections from global ionosphere maps.
Geometry-free linear combination This combination allows to estimate or remove geometry, including clocks and all non-dispersive effects in the signal. It contains ionospheric delays and all kind of bias that are frequency-dependents (hardware biases, cycle slips, and ambiguities). ' c[ dt dt ] r s T denotes the non-dispersive signal travel distance s r
Geometry-free linear combination dm I ) f f ( ρ P dm I ρ P δm N λ I ) f f -( ρ L δm N λ -I ρ L ' ' ' ' is contamineted by receiver and satellite biases are contamineted by multipath dm and i I δm i
Geometry-free linear combination ' ) ( ) ( N N I f f f f PD PD L L ' 4,9 5,9 3,9 4,9,546,546,546,546 PD PD L L N N I
Geometry-free linear combination ),,, ( diag b L 4,663 33,48 6,97 4,94,3 4,779 9,86 5,94 6,34 8,87 sim par / 6,53 ; / 5,8 ;,86 ;,978 ' N N I
Geometry-free linear combination It can also be written as: It removes geometry, including clocks, and all non-dispersive effects It contains ionospheric delays and all kind of bias that are frequencydependents (hardware biases, cycle slips, and ambiguities).
Geometry free (considering the biases) provide STEC It can be used for batch or recursive approach
Melbourne-Wübbena linear combination A very useful combination is the Melbourne-Wübbena (MW) combination, which is composed by phase and code measurements. (WL carrier and NL Pseudorange). L MW = f f f L f L f + f f P + f P = L W + P N The MW combination fulfills the conditions for geometry free and ionosphere-free combinations. MW combination yields a biased estimate of the wide-lane ambiguity
The NL and WL combinations introduced here have been confined to using signals from only two frequencies. However, narrow-lane, intermediate-lane, or wide-lane combinations can of course also be formed by using observations on three or more frequencies simultaneously. A variety of combinations with different characteristics with respect to noise amplification and suppression of ionospheric delays has been introduced in the literature
Multipath combination Multipath effects are mostly considered a nuisance in GNSS measurements and it is important to have an understanding of the magnitude of this error on the observations.
Combination of Multisatellite and Multireceiver All combinations presented so far, may be combined further with different receivers and satellites Between receivers single differences Between satellites single differences Double difference. Triple difference.
Between receivers single differences Assuming time-synchronized measurements of the same satellite k from two receivers and P k, # P k # P k # # means that we can apply such equation for any of the previous combination presented Like Iono free, WL, NL or others.
The satellite clock offset, which is identical for two timesynchronized observations at different receivers irrespective of the antenna distance, has dropped out. The tropospheric, ionospheric and group delay variations are still present in the equations, but strongly correlate with the distance of the two antennas. For close antennas, they reduce and can be disregarded. k k k k k k P,, cdt cd T I e k k k, cd k d rj d s j have been retained The reason for this is that only if receivers with identical correlators are used in the single difference, the combined satellite and receiver bias can be split up into the individual terms
Between receivers single differences k k k k k k k k k L,, cdt, c, T, I, ( N, w, ),
Between satellites single differences l k l k l k l k l k l k l k e I T cd cdt P,,,,,,, l k k k l l l l k l k l k l k l k l k w N w N I T c cdt L,,,,,,, ) ( ) ( The contribution of the system-time offset is contained in the differential satellite clock offset l k cd,
Double Difference (DD) DD can be formed with observations from a pair of receivers and and a pair of satellites k and l P cd T k, l k, l k, l k, l k, l,,,,, I e k, l, L k, l k, l k, l k, l k, l k, l k, l,, c, T, I, N, w, ) ( k, l,
P cd T k, l k, l k, l k, l k, l,,,,, I e k, l, The bias double differencesare present unless receivers with identical front-end characteristics and correlator settings are used L k, l k, l k, l k, l k, l k, l k, l,, c, T, I, N, w, ) ( k, l, The DD carrier-phase biasesare retained in the equations and will only drop out if receivers with compatible frontend and correlator design are used
Trimple Difference (TD) Forming SD and DD of observations eliminates some nuisance parameters, especially on short baselines where spatial correlation of signals delays can be exploited. Another method to eliminate errors is to take advantage of temporal correlation and make the differences at different epochs. All constant parameters will be eliminates.
Obtaining CM of the DD observables Epoch i l T i n n [ L, L,..., L, L, L,..., L ] l I i n being the variance of the considered observatio n (or combinatio n).
l [ I I ] l i SD n n i l I n SDi
l Cl DDi SD i For n satellites we have n- independent DDs and n(n-)/ possible DDs. C............ DD i............... C......... DDi............
DD CM for n epochs ( stations) It will be a block diagonal matrix, since no correlations are assumed between epochs
DD CM for a network ( epoch) DD i T T [ ] [ CC ] Kronecker product For a network defined by the following baselines: -; -3, 3-4;...;(m-)-m, will be given by:............ For m baselines we have m- independent baselines and m(m-)/ possible baselines.
Exercise Consider 3 stations and 8 GPS satellites being tracked simultaneously from dual frequency receivers at these stations. How many Ion-free DDs are possible for form at each baseline and in the network (dependent and independent observations) for both carrier and pseudorange?
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